176. CMB 2002 (vol 45 pp. 265)
 Nawrocki, Marek

On the Smirnov Class Defined by the Maximal Function
H.~O.~Kim has shown that contrary to the case of
$H^p$space, the Smirnov class $M$ defined by the radial maximal
function is essentially smaller than the classical Smirnov class
of the disk. In the paper we show that these two classes have the
same corresponding locally convex structure, {\it i.e.} they have the
same dual spaces and the same Fr\'echet envelopes. We describe a
general form of a continuous linear functional on $M$ and
multiplier from $M$ into $H^p$, $0 < p \leq \infty$.
Keywords:Smirnov class, maximal radial function, multipliers, dual space, FrÃ©chet envelope Categories:46E10, 30A78, 30A76 

177. CMB 2002 (vol 45 pp. 309)
 Xia, Jingbo

Joint Mean Oscillation and Local Ideals in the Toeplitz Algebra II: Local Commutivity and Essential Commutant
A wellknown theorem of Sarason [11] asserts that if $[T_f,T_h]$ is
compact for every $h \in H^\infty$, then $f \in H^\infty + C(T)$.
Using local analysis in the full Toeplitz algebra $\calT = \calT
(L^\infty)$, we show that the membership $f \in H^\infty + C(T)$ can
be inferred from the compactness of a much smaller collection of
commutators $[T_f,T_h]$. Using this strengthened result and a theorem
of Davidson [2], we construct a proper $C^\ast$subalgebra $\calT
(\calL)$ of $\calT$ which has the same essential commutant as that of
$\calT$. Thus the image of $\calT (\calL)$ in the Calkin algebra does
not satisfy the double commutant relation [12], [1]. We will also
show that no {\it separable} subalgebra $\calS$ of $\calT$ is capable
of conferring the membership $f \in H^\infty + C(T)$ through the
compactness of the commutators $\{[T_f,S] : S \in \calS\}$.
Categories:46H10, 47B35, 47C05 

178. CMB 2002 (vol 45 pp. 232)
 Ji, Min; Shen, Zhongmin

On Strongly Convex Indicatrices in Minkowski Geometry
The geometry of indicatrices is the foundation of Minkowski geometry.
A strongly convex indicatrix in a vector space is a strongly convex
hypersurface. It admits a Riemannian metric and has a distinguished
invariant(Cartan) torsion. We prove the existence of nontrivial
strongly convex indicatrices with vanishing mean torsion and discuss
the relationship between the mean torsion and the Riemannian curvature
tensor for indicatrices of Randers type.
Categories:46B20, 53C21, 53A55, 52A20, 53B40, 53A35 

179. CMB 2002 (vol 45 pp. 3)
 Azagra, D.; Dobrowolski, T.

RealAnalytic Negligibility of Points and Subspaces in Banach Spaces, with Applications
We prove that every infinitedimensional Banach space $X$ having a
(not necessarily equivalent) realanalytic norm is realanalytic
diffeomorphic to $X \setminus \{0\}$. More generally, if $X$ is an
infinitedimensional Banach space and $F$ is a closed subspace of $X$
such that there is a realanalytic seminorm on $X$ whose set of zeros
is $F$, and $X/F$ is infinitedimensional, then $X$ and $X \setminus
F$ are realanalytic diffeomorphic. As an application we show the
existence of realanalytic free actions of the circle and the
$n$torus on certain Banach spaces.
Categories:46B20, 58B99 

180. CMB 2002 (vol 45 pp. 60)
181. CMB 2002 (vol 45 pp. 46)
182. CMB 2001 (vol 44 pp. 504)
 Zhang, Yong

Weak Amenability of a Class of Banach Algebras
We show that, if a Banach algebra $\A$ is a left ideal in its second
dual algebra and has a left bounded approximate identity, then the
weak amenability of $\A$ implies the ($2m+1$)weak amenability of $\A$
for all $m\geq 1$.
Keywords:$n$weak amenability, left ideals, left bounded approximate identity Categories:46H20, 46H10, 46H25 

183. CMB 2001 (vol 44 pp. 355)
 Weaver, Nik

Hilbert Bimodules with Involution
We examine Hilbert bimodules which possess a (generally unbounded)
involution. Topics considered include a linking algebra
representation, duality, locality, and the role of these bimodules
in noncommutative differential geometry
Categories:46L08, 46L57, 46L87 

184. CMB 2001 (vol 44 pp. 370)
 Weston, Anthony

On Locating Isometric $\ell_{1}^{(n)}$
Motivated by a question of Per Enflo, we develop a hypercube criterion
for locating linear isometric copies of $\lone$ in an arbitrary real
normed space $X$.
The said criterion involves finding $2^{n}$ points in $X$ that satisfy
one metric equality. This contrasts nicely to the standard classical
criterion wherein one seeks $n$ points that satisfy $2^{n1}$ metric
equalities.
Keywords:normed spaces, hypercubes Categories:46B04, 05C10, 05B99 

185. CMB 2001 (vol 44 pp. 335)
 Stacey, P. J.

Inductive Limit Toral Automorphisms of Irrational Rotation Algebras
Irrational rotation $C^*$algebras have an inductive limit
decomposition in terms of matrix algebras over the space of continuous
functions on the circle and this decomposition can be chosen to be
invariant under the flip automorphism. It is shown that the flip is
essentially the only toral automorphism with this property.
Categories:46L40, 46L35 

186. CMB 2001 (vol 44 pp. 105)
187. CMB 2000 (vol 43 pp. 418)
 Gong, Guihua; Jiang, Xinhui; Su, Hongbing

Obstructions to $\mathcal{Z}$Stability for Unital Simple $C^*$Algebras
Let $\cZ$ be the unital simple nuclear infinite dimensional
$C^*$algebra which has the same Elliott invariant as $\bbC$,
introduced in \cite{JS}. A $C^*$algebra is called $\cZ$stable
if $A \cong A \otimes \cZ$. In this note we give some necessary
conditions for a unital simple $C^*$algebra to be $\cZ$stable.
Keywords:simple $C^*$algebra, $\mathcal{Z}$stability, weak (un)perforation in $K_0$ group, property $\Gamma$, finiteness Category:46L05 

188. CMB 2000 (vol 43 pp. 320)
189. CMB 2000 (vol 43 pp. 368)
 Litvak, A. E.

KahaneKhinchin's Inequality for QuasiNorms
We extend the recent results of R.~Lata{\l}a and O.~Gu\'edon about
equivalence of $L_q$norms of logconcave random variables
(KahaneKhinchin's inequality) to the quasiconvex case. We
construct examples of quasiconvex bodies $K_n \subset \R$ which
demonstrate that this equivalence fails for uniformly distributed
vector on $K_n$ (recall that the uniformly distributed vector on a
convex body is logconcave). Our examples also show the lack of the
exponential decay of the ``tail" volume (for convex bodies such
decay was proved by M.~Gromov and V.~Milman).
Categories:46B09, 52A30, 60B11 

190. CMB 2000 (vol 43 pp. 257)
191. CMB 2000 (vol 43 pp. 138)
 Boyd, C.

Exponential Laws for the Nachbin Ported Topology
We show that for $U$ and $V$ balanced open subsets of (Qno) Fr\'echet
spaces $E$ and $F$ that we have the topological identity
$$
\bigl( {\cal H}(U\times V), \tau_\omega \bigr) = \biggl( {\cal H}
\Bigl( U; \bigl( {\cal H}(V), \tau_\omega \bigr) \Bigr), \tau_\omega
\biggr).
$$
Analogous results for the compact open topology have long been
established. We also give an example to show that the (Qno)
hypothesis on both $E$ and $F$ is necessary.
Categories:46G20, 18D15, 46M05 

192. CMB 2000 (vol 43 pp. 208)
 Matoušková, Eva

Extensions of Continuous and Lipschitz Functions
We show a result slightly more general than the following. Let $K$
be a compact Hausdorff space, $F$ a closed subset of $K$, and $d$ a
lower semicontinuous metric on $K$. Then each continuous function
$f$ on $F$ which is Lipschitz in $d$ admits a continuous extension on
$K$ which is Lipschitz in $d$. The extension has the same supremum
norm and the same Lipschitz constant.
As a corollary we get that a Banach space $X$ is reflexive if and only
if each bounded, weakly continuous and norm Lipschitz function
defined on a weakly closed subset of $X$ admits a weakly continuous,
norm Lipschitz extension defined on the entire space $X$.
Keywords:extension, continous, Lipschitz, Banach space Categories:54C20, 46B10 

193. CMB 2000 (vol 43 pp. 193)
 Magajna, Bojan

C$^*$Convexity and the Numerical Range
If $A$ is a prime C$^*$algebra, $a \in A$ and $\lambda$ is in the
numerical range $W(a)$ of $a$, then for each $\varepsilon > 0$ there
exists an element $h \in A$ such that $\norm{h} = 1$ and $\norm{h^*
(a\lambda)h} < \varepsilon$. If $\lambda$ is an extreme point of
$W(a)$, the same conclusion holds without the assumption that $A$ is
prime. Given any element $a$ in a von Neumann algebra (or in a
general C$^*$algebra) $A$, all normal elements in the weak* closure
(the norm closure, respectively) of the C$^*$convex hull of $a$ are
characterized.
Categories:47A12, 46L05, 46L10 

194. CMB 2000 (vol 43 pp. 69)
195. CMB 1999 (vol 42 pp. 274)
 Dădărlat, Marius; Eilers, Søren

The Bockstein Map is Necessary
We construct two nonisomorphic nuclear, stably finite,
real rank zero $C^\ast$algebras $E$ and $E'$ for which
there is an isomorphism of ordered groups
$\Theta\colon \bigoplus_{n \ge 0} K_\bullet(E;\ZZ/n) \to
\bigoplus_{n \ge 0} K_\bullet(E';\ZZ/n)$ which is compatible
with all the coefficient transformations. The $C^\ast$algebras
$E$ and $E'$ are not isomorphic since there is no $\Theta$
as above which is also compatible with the Bockstein operations.
By tensoring with Cuntz's algebra $\OO_\infty$ one obtains a pair
of nonisomorphic, real rank zero, purely infinite $C^\ast$algebras
with similar properties.
Keywords:$K$theory, torsion coefficients, natural transformations, Bockstein maps, $C^\ast$algebras, real rank zero, purely infinite, classification Categories:46L35, 46L80, 19K14 

196. CMB 1999 (vol 42 pp. 344)
197. CMB 1999 (vol 42 pp. 321)
 Kikuchi, Masato

Averaging Operators and Martingale Inequalities in Rearrangement Invariant Function Spaces
We shall study some connection between averaging operators and
martingale inequalities in rearrangement invariant function spaces.
In Section~2 the equivalence between Shimogaki's theorem and some
martingale inequalities will be established, and in Section~3 the
equivalence between Boyd's theorem and martingale inequalities with
change of probability measure will be established.
Keywords:martingale inequalities, rearrangement invariant function spaces Categories:60G44, 60G46, 46E30 

198. CMB 1999 (vol 42 pp. 221)
 Liu, Peide; Saksman, Eero; Tylli, HansOlav

Boundedness of the $q$MeanSquare Operator on VectorValued Analytic Martingales
We study boundedness properties of the $q$meansquare operator
$S^{(q)}$ on $E$valued analytic martingales, where $E$ is a
complex quasiBanach space and $2 \leq q < \infty$. We establish
that a.s. finiteness of $S^{(q)}$ for every bounded $E$valued
analytic martingale implies strong $(p,p)$type estimates for
$S^{(q)}$ and all $p\in (0,\infty)$. Our results yield new
characterizations (in terms of analytic and stochastic properties
of the function $S^{(q)}$) of the complex spaces $E$ that admit an
equivalent $q$uniformly PLconvex quasinorm. We also obtain a
vectorvalued extension (and a characterization) of part of an
observation due to Bourgain and Davis concerning the
$L^p$boundedness of the usual squarefunction on scalarvalued
analytic martingales.
Categories:46B20, 60G46 

199. CMB 1999 (vol 42 pp. 139)
200. CMB 1999 (vol 42 pp. 118)
 Rao, T. S. S. R. K.

Points of Weak$^\ast$Norm Continuity in the Unit Ball of the Space $\WC(K,X)^\ast$
For a compact Hausdorff space with a dense set of isolated points, we
give a complete description of points of weak$^\ast$norm continuity
in the dual unit ball of the space of Banach space valued functions
that are continuous when the range has the weak topology. As an
application we give a complete description of points of weaknorm
continuity of the unit ball of the space of vector measures when
the underlying Banach space has the RadonNikodym property.
Keywords:Points of weak$^\ast$norm continuity, space of vector valued weakly continuous functions, $M$ideals Categories:46B20, 46E40 
